reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem
  for n holds Sum(Rseq * seq, n+1) = (Rseq * Partial_Sums(seq)).(n+1) -
  Sum((Rseq^\1 - Rseq) * Partial_Sums(seq), n) by Th53;
